Nested Sequent Calculi for Conditional Logics

Regis Alenda1, Nicola Olivetti1, and Gian Luca Pozzato2

1 Aix-Marseille Universite, CNRS, LSIS UMR 7296, 13397, Marseille, France -{regis.alenda,nicola.olivetti}@univ-cezanne.fr

2 Dip. Informatica - Univ. di Torino - Italypozzato@di.unito.it

Abstract. Nested sequent calculi are a useful generalization of ordinary sequentcalculi, where sequents are allowed to occur within sequents. Nested sequent cal-culi have been profitably employed in the area of (multi)-modal logic to obtainanalytic and modular proof systems for these logics. In thiswork, we extend therealm of nested sequents by providing nested sequent calculi for the basic con-ditional logic CK and some of its significant extensions. Thecalculi are internal(a sequent can be directly translated into a formula), cut-free and analytic. More-over, they can be used to design (sometimes optimal) decision procedures for therespective logics, and to obtain complexity upper bounds. Our calculi are an ar-gument in favour of nested sequent calculi for modal logics and alike, showingtheir versatility and power.

1 IntroductionThe recent history of the conditional logics starts with thework by Lewis [16, 17], whoproposed them in order to formalize a kind of hypothetical reasoning (ifA were thecase thenB), that cannot be captured by classical logic with material implication. Oneoriginal motivation was to formalizecounterfactual sentences, i.e. conditionals of theform “if A were the case thenB would be the case”, whereA is false. Conditionallogics have found an interest in several fields of artificial intelligence and knowledgerepresentation. They have been used to reason about prototypical properties [10] andto model belief change [14, 12]. Moreover, conditional logics can provide an axiomaticfoundation of nonmonotonic reasoning [5, 15], here a conditionalA ⇒ B is read as “innormal circumstances ifA thenB”. Recently, a kind of (multi)-conditional logics [3, 4]have been used to formalize epistemic change in a multi-agent setting.

Semantically, all conditional logics enjoy a possible world semantics, with the in-tuition that a conditionalA ⇒ B is true in a worldx, if B is true in the set of worldswhereA is true and that are most similar/closest/“as normal as”x. Since there are dif-ferent ways of formalizing “the set of worlds similar/closest/...” to a given world, thereare expectedly rather different semantics for conditionallogics, from the most generalselection function semantics to the stronger sphere semantics.

From the point of view of proof-theory and automated deduction, conditional logicsdo not have however a state of the art comparable with, say, the one of modal logics,where there are well-established alternative calculi, whose proof-theoretical and com-putational properties are well-understood. This is partially due to the mentioned lackof a unifying semantics; as a matter of fact the most general semantics, theselectionfunction one, is of little help for proof-theory, and the preferential/sphere semanticsonly captures a subset of (actually rather strong) systems.Similarly to modal logics andother extensions/alternative to classical logics two types of calculi have been studied:externalcalculi which make use of labels and relations on them to import the semantics

into the syntax, andinternal calculi which stay within the language, so that a “con-figuration” (sequent, tableaux node...) can be directly interpreted as a formula of thelanguage. Just to mention some work, to first stream belongs [2] proposing a calculusfor (unnested) cumulative logicC (see below). More recently, [18] presents modularlabeled calculi (of optimal complexity) for CK and some of its extensions, basing onthe selection function semantics, and [13] presents modular labeled calculi for prefer-ential logic PCL and its extensions. The latter calculi takeadvantage of a sort of hybridmodal translation. To the second stream belong the calculi by Gent [9] and by de Swart[21] for Lewis’ logic VC and neighbours. These calculi manipulate set of formulas andprovide a decision procedure, although they comprise an infinite set of rules. Very re-cently, some internal calculi for CK and some extensions (with any combination of MP,ID, CEM) have been proposed by Pattinson and Schroder [19].The calculi are obtainedby a general method for closing a set of rules (correspondingto Hilbert axioms) withrespect to the cut rule. These calculi have optimal complexity; notice that some of therules do not have a fixed number of premises. These calculi have been extended to pref-erential conditional logics [20], i.e. including cumulativity (CM) and or-axiom (CA),although the resulting systems are fairly complicated.

In this paper we begin to investigatenested sequentscalculi for conditional logics.Nested sequents are a natural generalization of ordinary sequents where sequents are al-lowed to occur within sequents. However a nested sequent always corresponds to a for-mula of the language, so that we can think of the rules as operating “inside a formula”,combining subformulas rather than just combining outer occurrences of formulas as inordinary sequents3. Limiting to modal logics, nested calculi have been provided, amongothers, for modal logics by Brunnler [7, 6] and by Fitting [8].

In this paper we treat the basic normal conditional logic CK (its role is the sameas K in modal logic) and its extensions with ID and CEM. We alsoconsider theflatfragment (i.e., without nested conditionals) of CK+CSO+ID, coinciding with the logicof cumulativity C introduced in [15]. The calculi are rather natural, all rules have afixed number of premises. The completeness is established bycut-elimination, whosepeculiarity is that it must take into account the substitution of equivalent antecedents ofconditionals (a condition corresponding to normality). The calculi can be used to obtaina decision procedure for the respective logics by imposing some restrictions preventingredundant applications of rules. In all cases, we get a PSPACE upper bound, a boundthat for CK+ID and CK+CSO+ID is optimal (but not for CK+CEM that is known tobe CONP). For flat CK+CSO+ID = cumulative logicC we also get a PSPACE bound,we are not aware of better upper bound for this logic (although we may suspect that itis not optimal). We can see the present work as a further argument in favor of nestedsequents as a useful tool to provide natural, yet computationally adequate, calculi formodal extensions of classical logics.

Technical details and proofs can be found in the accompanying report [1].

2 Conditional LogicsA propositional conditional languageL contains: - a set of propositional variablesATM ; - the symbol offalse⊥; - a set of connectives⊤, ∧, ∨, ¬, →, ⇒. We define

3 In this sense, they are a special kind of “deep inference” calculi by Guglielmi and colleagues.

formulas ofL as follows: -⊥ and the propositional variables ofATM areatomic for-mulas; - if A andB are formulas, then¬A andA ⊗ B arecomplex formulas, where⊗ ∈ {∧,∨,→,⇒}. We adopt theselection function semantics, that we briefly recallhere. We consider a non-empty set of possible worldsW . Intuitively, the selection func-tion f selects, for a worldw and a formulaA, the set of worlds ofW which arecloserto w given the informationA. A conditional formulaA ⇒ B holds in a worldw if theformulaB holds inall the worlds selected byf for w andA.

Definition 1 (Selection function semantics).A model is a tripleM = 〈W , f, [ ]〉where: -W is a non empty set ofworlds; - f is theselection functionf : W × 2W −→2W ; - [ ] is theevaluation function, which assigns to an atomP ∈ ATM the set ofworlds whereP is true, and is extended to boolean formulas as usual, whereas forconditional formulas[A ⇒ B] = {w ∈ W | f(w, [A]) ⊆ [B]}.

We have definedf taking [A] rather thanA (i.e. f (w, [A]) rather thanf (w, A)) as anargument; this is equivalent to definef on formulas, i.e.f(w, A) but imposing that if[A] = [A

′

] in the model, thenf(w, A) = f(w, A′

). This condition is callednormality.The semantics above characterizes thebasic conditional system, called CK [17]. An

axiomatization of CK is given by (⊢ denotes provability in the axiom system):– any axiomatization of the classical propositional calculus;– If ⊢ A and⊢ A → B, then⊢ B (Modus Ponens)– If ⊢ A ↔ B then⊢ (A ⇒ C) ↔ (B ⇒ C) (RCEA)– If ⊢ (A1 ∧ · · · ∧ An) → B then⊢ (C ⇒ A1 ∧ · · · ∧ C ⇒ An) → (C ⇒ B) (RCK)

Other conditional systems are obtained by assuming furtherproperties on the selec-tion function; we consider the following standard extensions of the basic system CK:

System Axiom Model conditionID A ⇒ A f(w, [A]) ⊆ [A]

CEM (A ⇒ B) ∨ (A ⇒ ¬B) | f(w, [A]) |≤ 1

CSO(A ⇒ B) ∧ (B ⇒ A) →((A ⇒ C) → (B ⇒ C))

f(w, [A]) ⊆ [B] andf(w, [B]) ⊆ [A]impliesf(w, [A]) = f(w, [B])

The above axiomatization is complete with respect to the semantics [17].

3 Nested Sequent CalculiNS for Conditional LogicsIn this section we present nested sequent calculiNS, whereS is an abbreviation forCK+X, and X={CEM, ID, CEM+ID}. As usual, completeness is an easy consequenceof the admissibility of cut. We are also able to turnNS into a terminating calculus,which gives us a decision procedure for the respective conditional logics.

Definition 2. A nested sequentΓ is defined inductively as follows: - A finite multiset offormulas is a nested sequent. - IfA is a formula andΓ is a nested sequent, then[A : Γ ]is a nested sequent. - A finite multiset of nested sequents is anested sequent.

A nested sequent can be displayed as

A1, . . . , Am, [B1 : Γ1], . . . , [Bn : Γn],

wheren, m ≥ 0, A1, . . . , Am, B1, . . . , Bn are formulas andΓ1, . . . , Γn are nested se-quents. The depthd(Γ ) of a nested sequentΓ is defined as follows: - ifΓ = A1 . . . , An,

then d(Γ ) = 0; - if Γ = [A : ∆], thend(Γ ) = 1 + d(∆) - if Γ = Γ1, . . . , Γn,then d(Γ ) = max(Γi). A nested sequent can be directly interpreted as a formula,just replace “,” by∨ and “:” by ⇒. More explicitly, the interpretation of a nestedsequentA1, . . . , Am, [B1 : Γ1], . . . , [Bn : Γn] is inductively defined by the formulaF(Γ ) = A1 ∨ . . . ∨ Am ∨ (B1 ⇒ F(Γ1)) ∨ . . . ∨ (Bn ⇒ F(Γn)). For example, thenested sequentA, B, [A : C, [B : E, F ]], [A : D] denotes the formulaA ∨ B ∨ (A ⇒C ∨ (B ⇒ (E ∨ F ))) ∨ (A ⇒ D).

The specificity of nested sequent calculi is to allow inferences that apply withinformulas. In order to introduce the rules of the calculus, weneed the notion of context.Intuitively a context denotes a “hole”, auniqueempty position, within a sequent thatcan be filled by a formula/sequent. We use the symbol( ) to denote the empty context.A context is defined inductively as follows:

Definition 3. If ∆ is a nested sequent,Γ ( ) = ∆, ( ) is a context with depthd(Γ ( )) =0; if ∆ is a nested sequent andΣ( ) is a context,Γ ( ) = ∆, [A : Σ( )] is a context withdepthd(Γ ( )) = 1 + d(Σ( )).

Finally we define the result of filling “the hole” of a context by a sequent:

Definition 4. Let Γ ( ) be a context and∆ be a sequent, then the sequent obtained byfilling the context by∆, denoted byΓ (∆) is defined as follows: - ifΓ ( ) = Λ, ( ), thenΓ (∆) = Λ, ∆; - if Γ ( ) = Λ, [A : Σ( )], thenΓ (∆) = Λ, [A : Σ(∆)].

The calculiNS are shown in Figure 1. As usual, we say that a nested sequentΓ isderivablein NS if it admits aderivation. A derivation is a tree whose nodes are nestedsequents. A branch is a sequence of nodesΓ1, Γ2, . . . , Γn, . . . Each nodeΓi is obtainedfrom its immediate successorΓi−1 by applyingbackwarda rule ofNS, havingΓi−1

as the conclusion andΓi as one of its premises. A branch is closed if one of its nodesis an instance of axioms(AX) and(AX⊤), otherwise it is open. We say that a tree isclosed if all its branches are closed. A nested sequentΓ has a derivation inNS if thereis a closed tree havingΓ as a root. As an example, Figure 2 shows a derivation of (aninstance of) the axiom ID.

Γ(P,¬P ) (AX) (AX⊤)Γ(⊤)Γ(A ∧B) Γ(¬(A ∧B))

Γ(A ∨B) Γ(¬(A ∨B))

Γ(A)

Γ(¬A) Γ(¬B)

Γ(B)

Γ(A,B)

Γ(¬A,¬B)

Γ(A ⇒ B) Γ(¬(A ⇒ B), [A′ : ∆])

Γ(¬(A ⇒ B), [A′ : ∆,¬B])Γ([A : B]) A,¬A′

Γ([A : ∆])

Γ([A : ∆,¬A])

Γ([A : ∆], [B : Σ])

Γ([A : ∆,Σ], [B : Σ]) A,¬B B,¬A

(∧+) (∧−)

(∨−)(∨+)

(⇒+) (⇒−)

(ID) (CEM)

Γ(A→ B) Γ(¬(A→ B))

Γ(¬A,B) Γ(A)(→+) (→−)

(¬)

P ∈ ATM

Γ(A)

Γ(¬¬A)

A′,¬A

Γ(¬B)

Fig. 1. The nested sequent calculiNS.

The following lemma shows that axioms can be generalized to any formulaF :

Lemma 1. Given any formulaF , the sequentΓ (F,¬F ) is derivable inNS.

Fig. 2. A derivation of the axiom ID.

The easy proof is by induction on the complexity ofF .In [19] the authors propose optimal sequent calculi for CK and its extensions by any

combination of ID, MP and CEM. It is not difficult to see that the rulesCKg , CKIDg ,CKCEMg , CKCEMIDg of their calculi are derivable in our calculi.

3.1 Basic structural properties ofNS

First of all, we show that weakening and contraction are height-preserving admissiblein the calculiNS. Furthermore, we show that all the rules of the calculi, withthe excep-tions of (⇒−) and(CEM ), are height-preserving invertible. As usual, we define theheight of a derivation as the height of the tree corresponding to the derivation itself.

Lemma 2 (Admissibility of weakening).Weakening is height-preserving admissiblein NS: if Γ (∆) (resp.Γ ([A : ∆])) is derivable inNS with a derivation of heighth,then alsoΓ (∆, F ) (resp.Γ ([A : ∆, F ])) is derivable inNS with a proof of heighth′ ≤ h, whereF is either a formula of a nested sequent[B : Σ].

The easy proof is by induction on the height of the derivationof Γ (∆).

Lemma 3 (Invertibility). All the rules ofNS, with the exceptions of(⇒−) and(CEM ),are height-preserving invertible: ifΓ has a derivation of heighth and it is an instance ofa conclusion of a rule(R), then alsoΓi, i = 1, 2, are derivable inNS with derivationsof heightshi ≤ h, whereΓi are instances of the premises of(R).

Proof. Let us first consider the rule(ID). In this case, we can immediately conclude be-cause the premiseΓ ([A : ∆,¬A]) is obtained by weakening, which is height-preservingadmissible (Lemma 2), fromΓ ([A : ∆]). For the other rules, we proceed by inductionon the height of the derivation ofΓ . We only show the most interesting case of(⇒+).For the base case, consider(∗) Γ (A ⇒ B, ∆) where either (i)P ∈ ∆ and¬P ∈ ∆,i.e. (∗) is an instance of(AX), or (ii) ⊤ ∈ ∆, i.e. (∗) is an instance of(AX⊤); weimmediately conclude that alsoΓ ([A : B], ∆) is an instance of either(AX) in case(i) or (AX⊤) in case (ii).For the inductive step, we consider each rule ending (lookingforward) the derivation ofΓ (A ⇒ B). If the derivation is ended by an application of(⇒+) to Γ ([A : B]), we are done. Otherwise, we apply the inductive hypothesis to thepremise(s) and then we conclude by applying the same rule. �

It can be observed that a “weak” version of invertibility also holds for the rules(⇒−)and(CEM). Roughly speaking, ifΓ (¬(A ⇒ B), [A′ : ∆]), which is an instance of theconclusion of(⇒−), is derivable, then also the sequentΓ (¬(A ⇒ B), [A′ : ∆,¬B]),namely the left-most premise in the rule(⇒−), is derivable too. Similarly for(CEM).

Since the rules are invertible, it follows thatcontractionis admissible, that is to say:

Lemma 4 (Admissibility of contraction). Contraction is height-preserving admissi-ble inNS: if Γ (F, F ) has a derivation of heighth, then alsoΓ (F ) has a derivation ofheighth′ ≤ h, whereF is either a formula or a nested sequent[A : Σ].

3.2 Soundness of the calculiNS

To improve readability, we slightly abuse the notation identifying a sequentΓ with itsinterpreting formulaF(Γ ), thus we shall writeA ⇒ ∆, Γ ∧ ∆, etc. instead ofA ⇒F(Γ ),F(Γ ) ∧ F(∆). First of all we prove that nested inference in sound (similarly toBrunnler [7], Lemma 2.8).

Lemma 5. LetΓ ( ) be any context. If the formulaA1 ∧ . . . ∧ An → B, with n ≥ 0, is(CK+X)-valid, then alsoΓ (A1) ∧ . . . ∧ Γ (An) → Γ (B) is (CK+X) valid.

Proof. By induction on the depth of a contextΓ ( ). Let d(Γ ( )) = 0, thenΓ = Λ, ( ).SinceA1 ∧ . . . ∧ An → B is valid, by propositional reasoning, we have that also(Λ ∨ A1) ∧ . . . (Λ ∨ An) → (Λ ∨ B) is valid, that isΓ (A1) ∧ . . . ∧ Γ (An) → Γ (B)is valid. Letd(Γ ( )) > 0, thenΓ ( ) = ∆, [C : Σ( )]. By inductive hypothesis, wehave thatΣ(A1) ∧ . . . ∧ Σ(An) → Σ(B) is valid. By (RCK), we obtain that also(C ⇒ Σ(A1)) ∧ . . . ∧ (C ⇒ Σ(An)) → (C ⇒ Σ(B)) is valid. Then, we get that(Λ∨ (C ⇒ Σ(A1)))∧ . . .∧ (Λ∨ (C ⇒ Σ(An))) → (Λ∨ (C ⇒ Σ(B))) is also valid,that isΓ (A1) ∧ . . . ∧ Γ (An) → Γ (B) is valid. �

Theorem 1. If Γ is derivable inNS, thenΓ is valid.

Proof. By induction on the height of the derivation ofΓ . If Γ is an axiom, that isΓ = Γ (P,¬P ), then trivially P ∨ ¬P is valid; by Lemma 5 (casen = 0), we getΓ (P,¬P ) is valid. Similarly forΓ (⊤). OtherwiseΓ is obtained by a rule (R):- (R) is a propositional rule, sayΓ1, Γ2/∆, we first prove thatΓ1∧Γ2 → ∆ is valid. Allrules are easy, since for the empty context they are nothing else than trivial propositionaltautologies. We can then use Lemma 5 to propagate them to any context. For instance,let the rule (R) be (¬∨). Then(¬A ∧ ¬B) → ¬(A ∨ B) and, by the previous lemma,we get thatΓ (¬A)∧Γ (¬B) → Γ (¬(A∨B)). Thus ifΓ is derived by (R) from Γ1, Γ2,we use the inductive hypothesis thatΓ1 andΓ2 are valid and the above fact to conclude.- (R) is (⇒+): trivial by inductive hypothesis.- (R) is (⇒−) thenΓ = Γ (¬(A ⇒ B), [A′ : ∆]) is derived from(i) Γ (¬(A ⇒ B), [A′ :∆,¬B]), (ii) ¬A, A′, (iii) ¬A′, A. By inductive hypothesis we have thatA ↔ A′ isvalid. We show that also(∗) [¬(A ⇒ B)∨(A′ ⇒ (∆∨¬B))] → [¬(A ⇒ B)∨(A′ ⇒∆)] is valid, then we apply Lemma 5 and the inductive hypothesis to conclude. To prove(*), by (RCK) we have that the following is valid:[(A′ ⇒ B) ∧ (A′ ⇒ (∆ ∨ ¬B))] →(A′ ⇒ ∆). SinceA ↔ A′ is valid, by (RCEA) we get that(A ⇒ B) → (A′ ⇒ B) isvalid, so that also(A ⇒ B) → ((A′ ⇒ (∆ ∨ ¬B)) → (A′ ⇒ ∆)) is valid, then weconclude by propositional reasoning.- (R) is (ID), thenΓ = Γ ([A : ∆]) is derived fromΓ ([A : ∆,¬A]). We show that(A ⇒ (∆ ∨ ¬A)) → (A ⇒ ∆) is valid in CK+ID, then we conclude by Lemma 5 andby the inductive hypothesis. The mentioned formula is derivable: by (RCK) we obtain(A ⇒ A) → ((A ⇒ (∆ ∨ ¬A)) → (A ⇒ ∆)) so that we conclude by (ID).

- (R) is (CEM), thusΓ = Γ ([A : ∆], [A′, Σ]) and it is derived from(i) Γ ([A :∆, Σ], [A′ : Σ]), (ii) ¬A, A′, (iii) ¬A′, A. By inductive hypothesisA ↔ A′ is valid.We first show that(∗∗) (A ⇒ (∆ ∨ Σ)) → ((A ⇒ ∆) ∨ (A′ ⇒ Σ)). Then we con-clude as before by Lemma 5 and inductive hypothesis. To prove(**), we notice thatthe following is derivable by (RCK):(A ⇒ (∆ ∨ Σ)) → [(A ⇒ ¬∆) → (A ⇒ Σ)].By (CEM), the following is valid:(A ⇒ ∆) ∨ (A ⇒ ¬∆). Thus we get that(A ⇒(∆ ∨ Σ)) → [(A ⇒ ∆) ∨ (A ⇒ Σ)] is valid. SinceA ↔ A′ is valid, (by RCEA) wehave that also(A ⇒ Σ) → (A′ ⇒ Σ) is valid, obtaining (**). �

3.3 Completeness of the calculiNS

Completeness is an easy consequence of the admissibility ofthe following rulecut:

Γ (F ) Γ (¬F )(cut)

Γ (∅)

whereF is a formula. The standard proof of admissibility of cut proceeds by a doubleinduction over the complexity ofF and the sum of the heights of the derivations of thetwo premises of(cut), in the sense that we replace one cut by one or several cuts onformulas of smaller complexity, or on sequents derived by shorter derivations.However,in NS the standard proof does not work in the following case, in which the cut formulaF is a conditional formulaA ⇒ B:

(1) Γ ([A : B], [A′ : ∆])(⇒+)

(3) Γ (A ⇒ B, [A′ : ∆])

(2) Γ (¬(A ⇒ B), [A′ : ∆, ¬B]) A,¬A′

A′,¬A

(⇒+)Γ (¬(A ⇒ B), [A′ : ∆])

(cut)Γ ([A′ : ∆])

Indeed, even if we apply the inductive hypothesis on the heights of the derivations ofthe premises to cut(2) and(3), obtaining (modulo weakening, which is admissible byLemma 2) a derivation of(2′) Γ ([A′ : ∆,¬B], [A′ : ∆]), we cannot apply the inductivehypothesis on the complexity of the cut formula to(2′) and(1′) Γ ([A : ∆, B], [A′ : ∆])(obtained from(1) again by weakening). Such an application would be needed in orderto obtain a proof ofΓ ([A′ : ∆], [A′ : ∆]) and then to concludeΓ ([A′ : ∆]) sincecontraction is admissible (Lemma 4).

In order to prove the admissibility of cut forNS, we proceed as follows. First, weshow that ifA,¬A′ andA′,¬A are derivable, then ifΓ ([A : ∆]) is derivable, thenΓ ([A′ : ∆]), obtained by replacing[A : ∆] with [A′ : ∆], is also derivable. We provethat cut is admissible by “splitting” the notion of cut in twopropositions:

Theorem 2. In NS, the following propositions hold:(A) If Γ (F ) and Γ (¬F ) arederivable, so isΓ (∅), i.e.(cut) is admissible inNS; (B) if (I) Γ ([A : ∆]), (II) A,¬A′

and (III) A′,¬A are derivable, thenΓ ([A′ : ∆]) is derivable.

Proof. The proof of both is by mutual induction. To make the structure of the inductionclear call:Cut(c, h) the property (A) for anyΓ and any formulaF of complexitycand such that the sum of the heights of derivation of the premises ish. Similarly callSub(c) the assertion that (B) holds for anyΓ and any formulaA of complexityc. Thenwe show that following facts:

(i) ∀h Cut(0, h)(ii) ∀c Cut(c, 0)(iii) ∀c′ < c Sub(c) → (∀c′ < c ∀h′ Cut(c′, h′) ∧ ∀h′ < h Cut(c, h′) →Cut(c, h))(iv) ∀h Cut(c, h) → Sub(c)

This will prove that∀c ∀hCut(c, h) and∀c Sub(c), that is (A) and (B) hold. Theproof of (iv) (that is thatSub(c) holds) in itself is by induction on the heighth of thederivation of the premise (I) of (B). To save space, we only present the most interestingcases.Inductive step for(A): we distinguish the following two cases:• (case 1) the last step ofoneof the two premises is obtained by a rule (R) in whichFis not the principal formula. This case is standard, we can permute(R) over the cut, i.e.we cut the premise(s) of (R) and then we apply (R) to the result of cut.• (case 2)F is the principal formula in the last step ofbothderivations of the premisesof the cut inference. There are seven subcases:F is introduced a) by(∧−) - (∧+), b) by(∨−) - (∨+), c) by(→−) - (→+), d) by(⇒−) - (⇒+), e) by(⇒−) - (ID), f) by (⇒−)- (CEM ), g) by (CEM ) - (ID). We only show the most interesting case d), where thederivation is as follows:

(1) Γ (¬(A ⇒ B), [A′ : ∆,¬B]) A,¬A′

A′,¬A

(⇒−)Γ (¬(A ⇒ B), [A′ : ∆])

(2) Γ ([A : B], [A′ : ∆])(⇒+)

(3) Γ (A ⇒ B, [A′ : ∆])(cut)

Γ ([A′ : ∆])

First of all, since we have proofs forA,¬A′ and forA′,¬A andcp(A) < cp(A ⇒ B),we can apply the inductive hypothesis for (B) to(2), obtaining a proof of(2′) Γ ([A′ :B], [A′ : ∆]). By Lemma 2, from(3) we obtain a proof of at most the same heightof (3′) Γ (A ⇒ B, [A′ : ∆,¬B]). We can then conclude as follows: we first applythe inductive hypothesis on the height for (A) to cut(1) and(3′), obtaining a deriva-tion of (4) Γ ([A′ : ∆,¬B]). By Lemma 2, we have also a derivation of(4′) Γ ([A′ :∆,¬B], [A′ : ∆]). Again by Lemma 2, from(2′) we obtain a derivation of(2′′) Γ ([A′ :∆, B], [A′ : ∆]). We then apply the inductive hypothesis on the complexity ofthe cutformula to cut(2′′) and(4′), obtaining a proof ofΓ ([A′ : ∆], [A′ : ∆]), from which weconclude since contraction is admissible (Lemma 4).Inductive step for(B) (that is statement (iv) of the induction): we have to consider allpossible rules ending (looking forward) the derivation ofΓ ([A : ∆]). We only showthe most interesting case, when(⇒−) is applied by using[A : ∆] as principal formula.The derivation ends as follows:

(1) Γ (¬(C ⇒ D), [A : ∆, ¬D]) (2) C,¬A (3) A,¬C

(⇒−)Γ (¬(C ⇒ D), [A : ∆])

We can apply the inductive hypothesis to(1) to obtain a derivation of(1′) Γ (¬(C ⇒D), [A′ : ∆,¬D]). Since weakening is admissible (Lemma 2), from(II) we obtain aderivation of(II ′) C, A,¬A′, from (III) we obtain a derivation of(III ′) A′,¬A,¬C.Again by weakening, from(2) and(3) we obtain derivations of(2′) C,¬A,¬A′ and(3′) A′, A,¬C, respectively. We apply the inductive hypothesis of (A) that is that cutholds for the formulaA (of a given complexityc) and conclude as follows:

(1′) Γ (¬(C ⇒ D), [A′ : ∆, ¬D])

(II′) C, A,¬A

′ (2′) C,¬A,¬A′

(cut)C,¬A

′

(III′) A

′,¬A,¬C (3′) A

′, A,¬C

(cut)A

′,¬C

(⇒−)Γ (¬(C ⇒ D), [A′ : ∆])

�Theorem 3 (Completeness ofNS). If Γ is valid, then it is derivable inNS.

Proof. We prove that the axioms are derivable and that the set of derivable formulas isclosed under (Modus Ponens), (RCEA), and (RCK). A derivation of an instance of IDhas been shown in Figure 2. Here is a derivation of an instanceof CEM:

(AX)[A : B,¬B]

(AX)A,¬A

(AX)¬A, A

(CEM)[A : B], [A : ¬B]

(⇒+)[A : B], A ⇒ ¬B

(⇒+)A ⇒ B, A ⇒ ¬B

(∨+)(A ⇒ B) ∨ (A ⇒ ¬B)

For (Modus Ponens), the proof is standard and is omitted to save space. For (RCEA),we have to show that ifA ↔ B is derivable, then also(A ⇒ C) ↔ (B ⇒ C) isderivable. As usual,A ↔ B is an abbreviation for(A → B) ∧ (B → A). SinceA ↔ B is derivable, and since(∧+) and(→+) are invertible (Lemma 3), we have aderivation forA → B, then for(1) ¬A, B, and forB → A, then for(2) A,¬B. Wederive(A ⇒ C) → (B ⇒ C) (the other half is symmetric) as follows:

(AX)¬(A ⇒ C), [B : C,¬C] (1) ¬A, B (2) A,¬B

(⇒−)¬(A ⇒ C), [B : C]

(⇒+)¬(A ⇒ C), B ⇒ C

(→+)(A ⇒ C) → (B ⇒ C)

For (RCK), suppose that we have a derivation inNS of (A1 ∧ . . . ∧ An) → B. Since(→+) is invertible (Lemma 3), we have also a derivation ofB,¬(A1 ∧ . . .∧An). Since(∧−) is also invertible, then we have a derivation ofB,¬A1, . . . ,¬An and, by weaken-ing (Lemma 2), of(1) ¬(C ⇒ A1), . . . ,¬(C ⇒ An), [C : B,¬A1,¬A2, . . . ,¬An],from which we conclude as follows:

(1) ¬(C ⇒ A1), . . . ,¬(C ⇒ An), [C : B,¬A1,¬A2, . . . ,¬An]. . .

(⇒−)¬(C ⇒ A1), . . . ,¬(C ⇒ An), [C : B,¬A1,¬A2]

(AX)C,¬C

(AX)¬C, C

(⇒−)¬(C ⇒ A1), . . . ,¬(C ⇒ An), [C : B,¬A1]

(AX)C,¬C

(AX)¬C, C

(⇒−)¬(C ⇒ A1), . . . ,¬(C ⇒ An), [C : B]

(∧−)¬(C ⇒ A1 ∧ . . . ∧ C ⇒ An), [C : B]

(⇒+)¬(C ⇒ A1 ∧ . . . ∧ C ⇒ An), C ⇒ B

(→+)(C ⇒ A1 ∧ . . . ∧ C ⇒ An) → (C ⇒ B)

�

3.4 Termination and complexity ofNS

The rules (⇒−), (CEM), and(ID) may be applied infinitely often. In order to obtain aterminating calculus, we have to put some restrictions on the application of these rules.Let us first consider the systems without CEM. We put the following restrictions:

– apply (⇒−) to Γ (¬(A ⇒ B), [A′ : ∆]) only if ¬B 6∈ ∆;– apply(ID) to Γ ([A : ∆]) only if ¬A 6∈ ∆.

These restrictions impose that (⇒−) is applied only once to each formula¬(A ⇒ B)with a context[A′ : ∆] in each branch, and that(ID) is applied only once to eachcontext[A : ∆] in each branch.

Theorem 4. The calculiNS with the termination restrictions is sound and completefor their respective logics.

Proof. We show that it is useless to apply the rules(⇒−) and (ID) without the re-strictions. We only present the case of(⇒−). Suppose it is applied twice onΓ ([A :∆], [B : Σ]) in a branch. Since(⇒−) is “weakly” invertible, we can assume, with-out loss of generality, that the two applications of(⇒−) are consecutive, starting fromΓ (¬(A ⇒ B), [A′ : ∆,¬B,¬B]). By Lemma 4 (contraction), we have a derivation ofΓ ([A : ∆, Σ], [B : Σ]), and we can conclude with a (single) application of(⇒−). �

The above restrictions ensure a terminating proof search for the nested sequents for CKand CK+ID, in particular:

Theorem 5. The calculiNCK andNCK + ID with the termination restrictions givea PSPACE decision procedure for their respective logics.

For the systems allowing CEM, we need a more sophisticated machinery4, that allowsus also to conclude that:

Theorem 6. The calculiNCK+CEM andNCK+CEM+ID with the terminationrestrictions give aPSPACE decision procedure for their respective logics.

It is worth noticing that our calculi match the PSPACE lower-bound of the logics CKand CK+ID, and are thus optimal with respect to these logics.On the contrary thecalculi for CK+CEM(+ID) are not optimal, since validity in these logics is known to bedecidable inCONP. In future work we shall try to devise an optimal decision procedureby adopting a suitable strategy.

4 A calculus for the flat fragment of CK+CSO+IDIn this section we show another application of nested sequents to give an analytic calcu-lus for the flat fragment, i.e. without nested conditionals⇒, of CK+CSO+ID. This logicis well-known and it corresponds to logicC, the logic ofcumulativity, the weakest sys-tem in the family of KLM logics [15]. Formulas are restrictedto boolean combinationsof propositional formulas and conditionalsA ⇒ B whereA andB are propositionals.A sequent has then the form:

A1, . . . , Am, [B1 : ∆1], . . . , [Bm : ∆m]4 The termination of the calculi with(CEM ) can be found in [1].

whereBi and ∆i are propositional. The logic has also an alternative semantics interms ofweak preferential models. The rules ofNCK + CSO + ID are those onesof NCK + ID (restricted to the flat fragment) where the rule(⇒−) is replaced by therule (CSO):

Γ,¬(C ⇒ D), [A : ∆,¬D] Γ,¬(C ⇒ D), [A : C] Γ,¬(C ⇒ D), [C : A](CSO)

Γ, ¬(C ⇒ D), [A : ∆]

A derivation of an instance of CSO is easy and left to the reader. More interestingly, inFigure 3 we give an example of derivation of the cumulative axiom ((A ⇒ B)∧ (A ⇒C)) → (A ∧ B) ⇒ C.

Fig. 3. A derivation of the cumulative axiom((A ⇒ B) ∧ (A ⇒ C)) → (A ∧ B) ⇒ C. Weomit the first propositional steps, and we letΣ = ¬(A ⇒ B),¬(A ⇒ C).

Definition 5. A sequentΓ is reducedif it has the formΓ = Λ, Π, [B1 : ∆1], . . . , [Bm :∆m], whereΛ is a multiset of literals andΠ is a multiset ofnegativeconditionals.

The following proposition is a kind ofdisjunctive propertyfor reduced sequents.

Proposition 1. LetΓ = Λ, Π, [B1 : ∆1], . . . , [Bm : ∆m] be reduced, ifΓ is derivablethen for somei, the sequentΛ, Π, [Bi : ∆i] is (height-preserving) derivable.

Proposition 2. LetΓ = Σ, [B1 : ∆1], . . . , [Bm : ∆m] be any sequent, ifΓ is derivablethen it can be (height-preserving) derived from somereducedsequentsΓi = Σj , [B1 :∆1], . . . , [Bm : ∆m]. Moreover all rules applied to deriveΓ from Γi are either propo-sitional rules or the rule(⇒+).

Proposition 2 can be proved by permuting (downwards) all theapplications of proposi-tional and(⇒+) rules.

Proposition 3. Let Γ = Σ, [A : ∆], [A : ∆] be derivable, thenΓ = Σ, [A : ∆] is(height-preserving) derivable.

Proof. By Proposition 2,Γ is height-preserving derivable from a set of reduced se-quentsΣi, [A : ∆], [A : ∆]. By Proposition 1, eachΣi, [A : ∆] is derivable; we thenobtainΣ, [A : ∆] by applying the same sequence of rules. �

Theorem 7. Contraction is admissible inNCK +CSO+ID: if Γ (F, F ) is derivable,thenΓ (F ) is (height-preserving) derivable, whereF is either a formula or a nestedsequent[A : Σ].

Proof. (Sketch) IfF is a formula the proof is essentially the same as the one for Lemma4 in NS. If F = [A : Σ] we apply Proposition 3. �

Observe that the standard inductive proof of contraction does not work in the caseF = [A : ∆], that is why we have obtained it by Proposition 3 which in turnis basedon Proposition 1, a kind of disjunctive property. The same argumentdoes not extendimmediatly to the full language with nested conditionals.

As usual, we obtain completeness by cut-elimnation. As in case ofNS, the proofis by mutual induction together with a substitution property and is left to the reader dueto space limitations.

Theorem 8. In NCK + CSO + ID, the following propositions hold:(A) If Γ (F )andΓ (¬F ) are derivable, then so isΓ ; (B) if (I) Γ ([A : ∆]), (II) Γ ([A : A′]), and(III) Γ ([A′ : A]) are derivable, then so isΓ ([A′ : ∆]).

Theorem 9. The calculusNCK + CSO + ID is sound and complete for the flatfragment ofCK+CSO+ID.

Proof. For soundness just check the validity of the(CSO) rule. For completeness, onecan derive all instances of CSO axioms. Moreover the rules RCK and RCEA are deriv-able too (by using the rules(ID) and (CSO)). For closure under modus ponens, asusual we use the previous Theorem 8. Details are left to the reader. �

Termination of this calculus can be proved similarly to Theorem 5, details will be givenin a full version of the paper.

Theorem 10. The calculusNCK + CSO + ID with the termination restrictions givea PSPACE decision procedure for the flat fragment ofCK+CSO+ID.

We do not know whether this bound is optimal. The study of the optimal complexity forCK+CSO+ID is still open. A NEXP tableau calculus for cumulative logicC has beenproposed in [11]. In [20] the authors provide calculi for full (i.e. with nested condition-als) CK+ID+CM and CK+ID+CM+CA: these logics are related to CSO, but they donot concide with it, even for the flat fragment as CSO = CM+RT (restricted transitivity).Their calculi are internal, but rather complex as the make use of ingenious but highlycombinatorial rules. They obtain a PSPACE bound in all cases.

5 Conclusions and Future WorksIn this work we have provided nested sequent calculi for the basic normal conditionallogic and a few extensions of it. The calculi are analytic andtheir completeness isestablished via cut-elimination. The calculi can be used toobtain a decision procedure,in some cases of optimal complexity. We have also provided a nested sequent calculusfor the flat fragment of CK+CSO+ID, corresponding to the cumulative logicC of the

KLM framework. Even if for some of the logics considered in this paper there existother proof systems, we think that nested calculi are particularly natural internal calculi.Obviously, it is our goal to extent them to a wider spectrum ofconditional logics, inparticular preferential conditional logics, which still lack “natural” and internal calculi.We also intend to study improvements of the calculi towards efficiency, based on abetter control of duplication. Finally, we wish to take advantage of the calculi to studylogical properties of the corresponding systems (disjunction property, interpolation) ina constructive way.

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